with \(\mathcal {S}\) (0) the astrophysical \(S\)  factor near zero energy. By comparing Eq. REF  and Eq. REF , the values of \(\mathcal {S}\) (0) and
and it's second derivative \(\ddot{\mathcal {S}}\) (0) (wrt velocity \(v\) ) can be obtained by assuming a negligible first derivative \(\dot{\mathcal {S}}\) (0).
Thus, the direct-capture cross section for neutron-induced reaction can be calculated by [1]}, [2]},
\(\sigma =\frac{\mathcal {S}(0)}{v}=\frac{\mathcal {S}(0)+\frac{1}{2}\ddot{\mathcal {S}}(0)E}{v}.\) 
